Abstract

One of the most important achievements of the last century is the knowledge of the existence of non-countable sets. The proof by Cantor’s diagonal method requires the assumption of actual infinity. By two paradoxes we show that this method sometimes proves nothing because of it can involve self-referential definitions. To avoid this inconvenient, we introduce another proving method based upon the information in the involved object definitions. We also introduce the concepts of indiscernible mathematical construction and sub-cardinal. In addition, we show that the existence of indiscernible mathematical constructions is an unavoidable consequence of uncountability.

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References

Chaitin, Gregory (2012). How much information can there be in a real number? [reprint of mr2346819]. In: Computation, physics and beyond. Vol. 7160 of Lecture Notes in Comput. Sci.. pp. 247–251. Springer, Heidelberg.